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+The actual trajectory, along which a dynamical system may move from
+one point to another within a specified time, is derived by finding
+the path which minimizes the time integral of the difference between
-<
+the kinetic, $K$, and potential energies, $U$ \cite{tolman79}.
->
+the kinetic, $K$, and potential energies, $U$ \cite{Tolman1979}.
+\begin{equation}
+\delta \int_{t_1 }^{t_2 } {(K - U)dt = 0} ,
+\label{introEquation:halmitonianPrinciple1}
+Eq.~\ref{introEquation:motionHamiltonianCoordinate} and
+Eq.~\ref{introEquation:motionHamiltonianMomentum} are Hamilton's
+equation of motion. Due to their symmetrical formula, they are also
-<
+known as the canonical equations of motions \cite{Goldstein01}.
->
+known as the canonical equations of motions \cite{Goldstein2001}.
+An important difference between Lagrangian approach and the
+Hamiltonian approach is that the Lagrangian is considered to be a
+appropriate for application to statistical mechanics and quantum
+mechanics, since it treats the coordinate and its time derivative as
+independent variables and it only works with 1st-order differential
-<
+equations\cite{Marion90}.
->
+equations\cite{Marion1990}.
+In Newtonian Mechanics, a system described by conservative forces
+conserves the total energy \ref{introEquation:energyConservation}.
+isolated and conserve energy, Microcanonical ensemble(NVE) has a
+partition function like,
+\begin{equation}
-<
+\Omega (N,V,E) = e^{\beta TS}
-<
+\label{introEqaution:NVEPartition}.
->
+\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}.
+\end{equation}
+A canonical ensemble(NVT)is an ensemble of systems, each of which
+can share its energy with a large heat reservoir. The distribution
+many-body system in Statistical Mechanics. Fortunately, Ergodic
+Hypothesis is proposed to make a connection between time average and
+ensemble average. It states that time average and average over the
-<
+statistical ensemble are identical \cite{Frenkel1996, leach01:mm}.
->
+statistical ensemble are identical \cite{Frenkel1996, Leach2001}.
+\begin{equation}
+\langle A(q , p) \rangle_t = \mathop {\lim }\limits_{t \to \infty }
+\frac{1}{t}\int\limits_0^t {A(q(t),p(t))dt = \int\limits_\Gamma
+a properly weighted statistical average. This allows the researcher
+freedom of choice when deciding how best to measure a given
+observable. In case an ensemble averaged approach sounds most
-<
+reasonable, the Monte Carlo techniques\cite{metropolis:1949} can be
->
+reasonable, the Monte Carlo techniques\cite{Metropolis1949} can be
+utilized. Or if the system lends itself to a time averaging
+approach, the Molecular Dynamics techniques in
+Sec.~\ref{introSection:molecularDynamics} will be the best
+within the equations. Since 1990, geometric integrators, which
+preserve various phase-flow invariants such as symplectic structure,
+volume and time reversal symmetry, are developed to address this
-<
+issue. The velocity verlet method, which happens to be a simple
-<
+example of symplectic integrator, continues to gain its popularity
-<
+in molecular dynamics community. This fact can be partly explained
-<
+by its geometric nature.
->
+issue\cite{Dullweber1997, McLachlan1998, Leimkuhler1999}. The
->
+velocity verlet method, which happens to be a simple example of
->
+symplectic integrator, continues to gain its popularity in molecular
->
+dynamics community. This fact can be partly explained by its
->
+geometric nature.
+\subsection{\label{introSection:symplecticManifold}Symplectic Manifold}
+A \emph{manifold} is an abstract mathematical space. It locally
+\end{equation}In this case, $f$ is
+called a \emph{Hamiltonian vector field}.
-<
+Another generalization of Hamiltonian dynamics is Poisson Dynamics,
->
+Another generalization of Hamiltonian dynamics is Poisson
->
+Dynamics\cite{Olver1986},
+\begin{equation}
+\dot x = J(x)\nabla _x H \label{introEquation:poissonHamiltonian}
+\end{equation}
+The most obvious change being that matrix $J$ now depends on $x$.
-–
+The free rigid body is an example of Poisson system (actually a
-–
+Lie-Poisson system) with Hamiltonian function of angular kinetic
-–
+energy.
-–
+\begin{equation}
-–
+J(\pi ) = \left( {\begin{array}{*{20}c}
-–
+& {\pi _3 } & { - \pi _2 }  \\
-–
+   { - \pi _3 } & 0 & {\pi _1 }  \\
-–
+   {\pi _2 } & { - \pi _1 } & 0  \\
-–
+\end{array}} \right)
-–
+\end{equation}
-–
+\begin{equation}
-–
+H = \frac{1}{2}\left( {\frac{{\pi _1^2 }}{{I_1 }} + \frac{{\pi _2^2
-–
+}}{{I_2 }} + \frac{{\pi _3^2 }}{{I_3 }}} \right)
-–
+\end{equation}
-–
+\subsection{\label{introSection:exactFlow}Exact Flow}
+Let $x(t)$ be the exact solution of the ODE system,
+\subsection{\label{introSection:geometricProperties}Geometric Properties}
-<
+The hidden geometric properties of ODE and its flow play important
-<
+roles in numerical studies. Many of them can be found in systems
-<
+which occur naturally in applications.
->
+The hidden geometric properties\cite{Budd1999, Marsden1998} of ODE
->
+and its flow play important roles in numerical studies. Many of them
->
+can be found in systems which occur naturally in applications.
+Let $\varphi$ be the flow of Hamiltonian vector field, $\varphi$ is
+a \emph{symplectic} flow if it satisfies,
+The vector filed $f$ has reversing symmetry $h$ if $f = - \tilde f$.
+In other words, the flow of this vector field is reversible if and
+only if $ h \circ \varphi ^{ - 1}  = \varphi  \circ h $.
-+
-+
+A \emph{first integral}, or conserved quantity of a general
-+
+differential function is a function $ G:R^{2d}  \to R^d $ which is
-+
+constant for all solutions of the ODE $\frac{{dx}}{{dt}} = f(x)$ ,
-+
+\[
-+
+\frac{{dG(x(t))}}{{dt}} = 0.
-+
+\]
-+
+Using chain rule, one may obtain,
-+
+\[
-+
+\sum\limits_i {\frac{{dG}}{{dx_i }}} f_i (x) = f \bullet \nabla G,
-+
+\]
-+
+which is the condition for conserving \emph{first integral}. For a
-+
+canonical Hamiltonian system, the time evolution of an arbitrary
-+
+smooth function $G$ is given by,
-+
+\begin{eqnarray}
-+
+\frac{{dG(x(t))}}{{dt}} & = & [\nabla _x G(x(t))]^T \dot x(t) \\
-+
+                        & = & [\nabla _x G(x(t))]^T J\nabla _x H(x(t)). \\
-+
+\label{introEquation:firstIntegral1}
-+
+\end{eqnarray}
-+
-+
-+
+Using poisson bracket notion, Equation
-+
+\ref{introEquation:firstIntegral1} can be rewritten as
-+
+\[
-+
+\frac{d}{{dt}}G(x(t)) = \left\{ {G,H} \right\}(x(t)).
-+
+\]
-+
+Therefore, the sufficient condition for $G$ to be the \emph{first
-+
+integral} of a Hamiltonian system is
-+
+\[
-+
+\left\{ {G,H} \right\} = 0.
-+
+\]
-+
+As well known, the Hamiltonian (or energy) H of a Hamiltonian system
-+
+is a \emph{first integral}, which is due to the fact $\{ H,H\}  =
-+
+$.
-+
+When designing any numerical methods, one should always try to
+preserve the structural properties of the original ODE and its flow.
+\item Splitting methods
+\end{enumerate}
-<
+Generating function tends to lead to methods which are cumbersome
-<
+and difficult to use. In dissipative systems, variational methods
-<
+can capture the decay of energy accurately. Since their
-<
+geometrically unstable nature against non-Hamiltonian perturbations,
-<
+ordinary implicit Runge-Kutta methods are not suitable for
-<
+Hamiltonian system. Recently, various high-order explicit
-<
+Runge--Kutta methods have been developed to overcome this
->
+Generating function\cite{Channell1990} tends to lead to methods
->
+which are cumbersome and difficult to use. In dissipative systems,
->
+variational methods can capture the decay of energy
->
+accurately\cite{Kane2000}. Since their geometrically unstable nature
->
+against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta
->
+methods are not suitable for Hamiltonian system. Recently, various
->
+high-order explicit Runge-Kutta methods
->
+\cite{Owren1992,Chen2003}have been developed to overcome this
+instability. However, due to computational penalty involved in
+implementing the Runge-Kutta methods, they do not attract too much
+attention from Molecular Dynamics community. Instead, splitting have
+been widely accepted since they exploit natural decompositions of
-<
+the system\cite{Tuckerman92}.
->
+the system\cite{Tuckerman1992, McLachlan1998}.
+\subsubsection{\label{introSection:splittingMethod}Splitting Method}
+splitting gives a second-order decomposition,
+\begin{equation}
+\varphi _h  = \varphi _{1,h/2}  \circ \varphi _{2,h}  \circ \varphi
-<
+_{1,h/2} ,
-<
+\label{introEqaution:secondOrderSplitting}
->
+_{1,h/2} , \label{introEquation:secondOrderSplitting}
+\end{equation}
+which has a local error proportional to $h^3$. Sprang splitting's
+popularity in molecular simulation community attribute to its
+q(\Delta t) &= q(0) + \frac{{\Delta t}}{2}\left[ {\dot q(0) + \dot
+q(\Delta t)} \right]. %
-<
+ \label{introEquation:positionVerlet1}
->
+ \label{introEquation:positionVerlet2}
+\end{align}
+\subsubsection{\label{introSection:errorAnalysis}Error Analysis and Higher Order Methods}
+error of splitting method in terms of commutator of the
+operators(\ref{introEquation:exponentialOperator}) associated with
+the sub-flow. For operators $hX$ and $hY$ which are associate to
-<
+$\varphi_1(t)$ and $\varphi_2(t$ respectively , we have
->
+$\varphi_1(t)$ and $\varphi_2(t)$ respectively , we have
+\begin{equation}
+\exp (hX + hY) = \exp (hZ)
+\end{equation}
+\[
+[X,Y] = XY - YX .
+\]
-<
+Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we
-<
+can obtain
->
+Applying Baker-Campbell-Hausdorff formula\cite{Varadarajan1974} to
->
+Sprang splitting, we can obtain
+\begin{eqnarray*}
-<
+\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2
-<
+[X,Y]/4 + h^2 [Y,X]/4 \\ & & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\
-<
+& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3 [X,[X,Y]]/24 & & \mbox{} +
-<
+\ldots )
->
+\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\
->
+                                   &   & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\
->
+                                   &   & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots )
+\end{eqnarray*}
+Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local
+error of Spring splitting is proportional to $h^3$. The same
+\varphi _{b_m h}^2  \circ \varphi _{a_m h}^1  \circ \varphi _{b_{m -
+} h}^2  \circ  \ldots  \circ \varphi _{a_1 h}^1 .
+\end{equation}
-<
+Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher
->
+Careful choice of coefficient $a_1 \ldot b_m$ will lead to higher
+order method. Yoshida proposed an elegant way to compose higher
-<
+order methods based on symmetric splitting. Given a symmetric second
-<
+order base method $ \varphi _h^{(2)} $, a fourth-order symmetric
-<
+method can be constructed by composing,
->
+order methods based on symmetric splitting\cite{Yoshida1990}. Given
->
+a symmetric second order base method $ \varphi _h^{(2)} $, a
->
+fourth-order symmetric method can be constructed by composing,
+\[
+\varphi _h^{(4)}  = \varphi _{\alpha h}^{(2)}  \circ \varphi _{\beta
+h}^{(2)}  \circ \varphi _{\alpha h}^{(2)}
+\section{\label{introSection:molecularDynamics}Molecular Dynamics}
-<
+As a special discipline of molecular modeling, Molecular dynamics
-<
+has proven to be a powerful tool for studying the functions of
-<
+biological systems, providing structural, thermodynamic and
-<
+dynamical information.
->
+As one of the principal tools of molecular modeling, Molecular
->
+dynamics has proven to be a powerful tool for studying the functions
->
+of biological systems, providing structural, thermodynamic and
->
+dynamical information. The basic idea of molecular dynamics is that
->
+macroscopic properties are related to microscopic behavior and
->
+microscopic behavior can be calculated from the trajectories in
->
+simulations. For instance, instantaneous temperature of an
->
+Hamiltonian system of $N$ particle can be measured by
->
+\[
->
+T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}}
->
+\]
->
+where $m_i$ and $v_i$ are the mass and velocity of $i$th particle
->
+respectively, $f$ is the number of degrees of freedom, and $k_B$ is
->
+the boltzman constant.
-<
+\subsection{\label{introSec:mdInit}Initialization}
->
+A typical molecular dynamics run consists of three essential steps:
->
+\begin{enumerate}
->
+  \item Initialization
->
+    \begin{enumerate}
->
+    \item Preliminary preparation
->
+    \item Minimization
->
+    \item Heating
->
+    \item Equilibration
->
+    \end{enumerate}
->
+  \item Production
->
+  \item Analysis
->
+\end{enumerate}
->
+These three individual steps will be covered in the following
->
+sections. Sec.~\ref{introSec:initialSystemSettings} deals with the
->
+initialization of a simulation. Sec.~\ref{introSec:production} will
->
+discusses issues in production run. Sec.~\ref{introSection:Analysis}
->
+provides the theoretical tools for trajectory analysis.
-<
+\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion}
->
+\subsection{\label{introSec:initialSystemSettings}Initialization}
-<
+\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
->
+\subsubsection{Preliminary preparation}
-<
+A rigid body is a body in which the distance between any two given
-<
+points of a rigid body remains constant regardless of external
-<
+forces exerted on it. A rigid body therefore conserves its shape
-<
+during its motion.
->
+When selecting the starting structure of a molecule for molecular
->
+simulation, one may retrieve its Cartesian coordinates from public
->
+databases, such as RCSB Protein Data Bank \textit{etc}. Although
->
+thousands of crystal structures of molecules are discovered every
->
+year, many more remain unknown due to the difficulties of
->
+purification and crystallization. Even for the molecule with known
->
+structure, some important information is missing. For example, the
->
+missing hydrogen atom which acts as donor in hydrogen bonding must
->
+be added. Moreover, in order to include electrostatic interaction,
->
+one may need to specify the partial charges for individual atoms.
->
+Under some circumstances, we may even need to prepare the system in
->
+a special setup. For instance, when studying transport phenomenon in
->
+membrane system, we may prepare the lipids in bilayer structure
->
+instead of placing lipids randomly in solvent, since we are not
->
+interested in self-aggregation and it takes a long time to happen.
-<
+Applications of dynamics of rigid bodies.
->
+\subsubsection{Minimization}
-<
+\subsection{\label{introSection:lieAlgebra}Lie Algebra}
-<
-<
+\subsection{\label{introSection:DLMMotionEquation}The Euler Equations of Rigid Body Motion}
-<
-<
+\subsection{\label{introSection:otherRBMotionEquation}Other Formulations for Rigid Body Motion}
-<
-<
+\section{\label{introSection:correlationFunctions}Correlation Functions}
-<
-<
+\section{\label{introSection:langevinDynamics}Langevin Dynamics}
-<
-<
+\subsection{\label{introSection:LDIntroduction}Introduction and application of Langevin Dynamics}
->
+It is quite possible that some of molecules in the system from
->
+preliminary preparation may be overlapped with each other. This
->
+close proximity leads to high potential energy which consequently
->
+jeopardizes any molecular dynamics simulations. To remove these
->
+steric overlaps, one typically performs energy minimization to find
->
+a more reasonable conformation. Several energy minimization methods
->
+have been developed to exploit the energy surface and to locate the
->
+local minimum. While converging slowly near the minimum, steepest
->
+descent method is extremely robust when systems are far from
->
+harmonic. Thus, it is often used to refine structure from
->
+crystallographic data. Relied on the gradient or hessian, advanced
->
+methods like conjugate gradient and Newton-Raphson converge rapidly
->
+to a local minimum, while become unstable if the energy surface is
->
+far from quadratic. Another factor must be taken into account, when
->
+choosing energy minimization method, is the size of the system.
->
+Steepest descent and conjugate gradient can deal with models of any
->
+size. Because of the limit of computation power to calculate hessian
->
+matrix and insufficient storage capacity to store them, most
->
+Newton-Raphson methods can not be used with very large models.
-<
+\subsection{\label{introSection:generalizedLangevinDynamics}Generalized Langevin Dynamics}
->
+\subsubsection{Heating}
-<
+\begin{equation}
-<
+H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
-<
+\label{introEquation:bathGLE}
-<
+\end{equation}
-<
+where $H_B$ is harmonic bath Hamiltonian,
-<
+\[
-<
+H_B  =\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
-<
+}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  w_\alpha ^2 } \right\}}
-<
+\]
-<
+and $\Delta U$ is bilinear system-bath coupling,
-<
+\[
-<
+\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
-<
+\]
-<
+Completing the square,
-<
+\[
-<
+H_B  + \Delta U = \sum\limits_{\alpha  = 1}^N {\left\{
-<
+{\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
-<
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
-<
+w_\alpha ^2 }}x} \right)^2 } \right\}}  - \sum\limits_{\alpha  =
-<
+}^N {\frac{{g_\alpha ^2 }}{{2m_\alpha  w_\alpha ^2 }}} x^2
-<
+\]
-<
+and putting it back into Eq.~\ref{introEquation:bathGLE},
-<
+\[
-<
+H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
-<
+{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
-<
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
-<
+w_\alpha ^2 }}x} \right)^2 } \right\}}
-<
+\]
-<
+where
-<
+\[
-<
+W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
-<
+}}{{2m_\alpha  w_\alpha ^2 }}} x^2
-<
+\]
-<
+Since the first two terms of the new Hamiltonian depend only on the
-<
+system coordinates, we can get the equations of motion for
-<
+Generalized Langevin Dynamics by Hamilton's equations
-<
+\ref{introEquation:motionHamiltonianCoordinate,
-<
+introEquation:motionHamiltonianMomentum},
-<
+\begin{align}
-<
+\dot p &=  - \frac{{\partial H}}{{\partial x}}
-<
+       &= m\ddot x
-<
+       &= - \frac{{\partial W(x)}}{{\partial x}} - \sum\limits_{\alpha  = 1}^N {g_\alpha  \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right)}
-<
+\label{introEquation:Lp5}
-<
+\end{align}
-<
+, and
-<
+\begin{align}
-<
+\dot p_\alpha   &=  - \frac{{\partial H}}{{\partial x_\alpha  }}
-<
+                &= m\ddot x_\alpha
-<
+                &= \- m_\alpha  w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha}}{{m_\alpha  w_\alpha ^2 }}x} \right)
-<
+\end{align}
->
+Typically, Heating is performed by assigning random velocities
->
+according to a Gaussian distribution for a temperature. Beginning at
->
+a lower temperature and gradually increasing the temperature by
->
+assigning greater random velocities, we end up with setting the
->
+temperature of the system to a final temperature at which the
->
+simulation will be conducted. In heating phase, we should also keep
->
+the system from drifting or rotating as a whole. Equivalently, the
->
+net linear momentum and angular momentum of the system should be
->
+shifted to zero.
-<
+\subsection{\label{introSection:laplaceTransform}The Laplace Transform}
->
+\subsubsection{Equilibration}
-<
+\[
-<
+L(x) = \int_0^\infty  {x(t)e^{ - pt} dt}
-<
+\]
->
+The purpose of equilibration is to allow the system to evolve
->
+spontaneously for a period of time and reach equilibrium. The
->
+procedure is continued until various statistical properties, such as
->
+temperature, pressure, energy, volume and other structural
->
+properties \textit{etc}, become independent of time. Strictly
->
+speaking, minimization and heating are not necessary, provided the
->
+equilibration process is long enough. However, these steps can serve
->
+as a means to arrive at an equilibrated structure in an effective
->
+way.
-<
+\[
-<
+L(x + y) = L(x) + L(y)
-<
+\]
->
+\subsection{\label{introSection:production}Production}
-<
+\[
-<
+L(ax) = aL(x)
-<
+\]
->
+Production run is the most important step of the simulation, in
->
+which the equilibrated structure is used as a starting point and the
->
+motions of the molecules are collected for later analysis. In order
->
+to capture the macroscopic properties of the system, the molecular
->
+dynamics simulation must be performed in correct and efficient way.
-<
+\[
-<
+L(\dot x) = pL(x) - px(0)
->
+The most expensive part of a molecular dynamics simulation is the
->
+calculation of non-bonded forces, such as van der Waals force and
->
+Coulombic forces \textit{etc}. For a system of $N$ particles, the
->
+complexity of the algorithm for pair-wise interactions is $O(N^2 )$,
->
+which making large simulations prohibitive in the absence of any
->
+computation saving techniques.
->
->
+A natural approach to avoid system size issue is to represent the
->
+bulk behavior by a finite number of the particles. However, this
->
+approach will suffer from the surface effect. To offset this,
->
+\textit{Periodic boundary condition} (see Fig.~\ref{introFig:pbc})
->
+is developed to simulate bulk properties with a relatively small
->
+number of particles. In this method, the simulation box is
->
+replicated throughout space to form an infinite lattice. During the
->
+simulation, when a particle moves in the primary cell, its image in
->
+other cells move in exactly the same direction with exactly the same
->
+orientation. Thus, as a particle leaves the primary cell, one of its
->
+images will enter through the opposite face.
->
+\begin{figure}
->
+\centering
->
+\includegraphics[width=\linewidth]{pbc.eps}
->
+\caption[An illustration of periodic boundary conditions]{A 2-D
->
+illustration of periodic boundary conditions. As one particle leaves
->
+the left of the simulation box, an image of it enters the right.}
->
+\label{introFig:pbc}
->
+\end{figure}
->
->
+%cutoff and minimum image convention
->
+Another important technique to improve the efficiency of force
->
+evaluation is to apply cutoff where particles farther than a
->
+predetermined distance, are not included in the calculation
->
+\cite{Frenkel1996}. The use of a cutoff radius will cause a
->
+discontinuity in the potential energy curve. Fortunately, one can
->
+shift the potential to ensure the potential curve go smoothly to
->
+zero at the cutoff radius. Cutoff strategy works pretty well for
->
+Lennard-Jones interaction because of its short range nature.
->
+However, simply truncating the electrostatic interaction with the
->
+use of cutoff has been shown to lead to severe artifacts in
->
+simulations. Ewald summation, in which the slowly conditionally
->
+convergent Coulomb potential is transformed into direct and
->
+reciprocal sums with rapid and absolute convergence, has proved to
->
+minimize the periodicity artifacts in liquid simulations. Taking the
->
+advantages of the fast Fourier transform (FFT) for calculating
->
+discrete Fourier transforms, the particle mesh-based
->
+methods\cite{Hockney1981,Shimada1993, Luty1994} are accelerated from
->
+$O(N^{3/2})$ to $O(N logN)$. An alternative approach is \emph{fast
->
+multipole method}\cite{Greengard1987, Greengard1994}, which treats
->
+Coulombic interaction exactly at short range, and approximate the
->
+potential at long range through multipolar expansion. In spite of
->
+their wide acceptances at the molecular simulation community, these
->
+two methods are hard to be implemented correctly and efficiently.
->
+Instead, we use a damped and charge-neutralized Coulomb potential
->
+method developed by Wolf and his coworkers\cite{Wolf1999}. The
->
+shifted Coulomb potential for particle $i$ and particle $j$ at
->
+distance $r_{rj}$ is given by:
->
+\begin{equation}
->
+V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha
->
+r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow
->
+R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha
->
+r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb}
->
+\end{equation}
->
+where $\alpha$ is the convergence parameter. Due to the lack of
->
+inherent periodicity and rapid convergence,this method is extremely
->
+efficient and easy to implement.
->
+\begin{figure}
->
+\centering
->
+\includegraphics[width=\linewidth]{shifted_coulomb.eps}
->
+\caption[An illustration of shifted Coulomb potential]{An
->
+illustration of shifted Coulomb potential.}
->
+\label{introFigure:shiftedCoulomb}
->
+\end{figure}
->
->
+%multiple time step
->
->
+\subsection{\label{introSection:Analysis} Analysis}
->
->
+Recently, advanced visualization technique are widely applied to
->
+monitor the motions of molecules. Although the dynamics of the
->
+system can be described qualitatively from animation, quantitative
->
+trajectory analysis are more appreciable. According to the
->
+principles of Statistical Mechanics,
->
+Sec.~\ref{introSection:statisticalMechanics}, one can compute
->
+thermodynamics properties, analyze fluctuations of structural
->
+parameters, and investigate time-dependent processes of the molecule
->
+from the trajectories.
->
->
+\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties}
->
->
+Thermodynamics properties, which can be expressed in terms of some
->
+function of the coordinates and momenta of all particles in the
->
+system, can be directly computed from molecular dynamics. The usual
->
+way to measure the pressure is based on virial theorem of Clausius
->
+which states that the virial is equal to $-3Nk_BT$. For a system
->
+with forces between particles, the total virial, $W$, contains the
->
+contribution from external pressure and interaction between the
->
+particles:
->
+\[
->
+W =  - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot
->
+f_{ij} } } \right\rangle
+\]
-+
+where $f_{ij}$ is the force between particle $i$ and $j$ at a
-+
+distance $r_{ij}$. Thus, the expression for the pressure is given
-+
+by:
-+
+\begin{equation}
-+
+P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i
-+
+< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle
-+
+\end{equation}
-+
+\subsubsection{\label{introSection:structuralProperties}Structural Properties}
-+
-+
+Structural Properties of a simple fluid can be described by a set of
-+
+distribution functions. Among these functions,\emph{pair
-+
+distribution function}, also known as \emph{radial distribution
-+
+function}, is of most fundamental importance to liquid-state theory.
-+
+Pair distribution function can be gathered by Fourier transforming
-+
+raw data from a series of neutron diffraction experiments and
-+
+integrating over the surface factor \cite{Powles1973}. The
-+
+experiment result can serve as a criterion to justify the
-+
+correctness of the theory. Moreover, various equilibrium
-+
+thermodynamic and structural properties can also be expressed in
-+
+terms of radial distribution function \cite{Allen1987}.
-+
-+
+A pair distribution functions $g(r)$ gives the probability that a
-+
+particle $i$ will be located at a distance $r$ from a another
-+
+particle $j$ in the system
+\[
-<
+L(\ddot x) = p^2 L(x) - px(0) - \dot x(0)
->
+g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j
->
+\ne i} {\delta (r - r_{ij} )} } } \right\rangle.
+\]
-+
+Note that the delta function can be replaced by a histogram in
-+
+computer simulation. Figure
-+
+\ref{introFigure:pairDistributionFunction} shows a typical pair
-+
+distribution function for the liquid argon system. The occurrence of
-+
+several peaks in the plot of $g(r)$ suggests that it is more likely
-+
+to find particles at certain radial values than at others. This is a
-+
+result of the attractive interaction at such distances. Because of
-+
+the strong repulsive forces at short distance, the probability of
-+
+locating particles at distances less than about 2.5{\AA} from each
-+
+other is essentially zero.
-+
+%\begin{figure}
-+
+%\centering
-+
+%\includegraphics[width=\linewidth]{pdf.eps}
-+
+%\caption[Pair distribution function for the liquid argon
-+
+%]{Pair distribution function for the liquid argon}
-+
+%\label{introFigure:pairDistributionFunction}
-+
+%\end{figure}
-+
-+
+\subsubsection{\label{introSection:timeDependentProperties}Time-dependent
-+
+Properties}
-+
-+
+Time-dependent properties are usually calculated using \emph{time
-+
+correlation function}, which correlates random variables $A$ and $B$
-+
+at two different time
-+
+\begin{equation}
-+
+C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle.
-+
+\label{introEquation:timeCorrelationFunction}
-+
+\end{equation}
-+
+If $A$ and $B$ refer to same variable, this kind of correlation
-+
+function is called \emph{auto correlation function}. One example of
-+
+auto correlation function is velocity auto-correlation function
-+
+which is directly related to transport properties of molecular
-+
+liquids:
+\[
-<
+L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right) = G(p)H(p)
->
+D = \frac{1}{3}\int\limits_0^\infty  {\left\langle {v(t) \cdot v(0)}
->
+\right\rangle } dt
+\]
-+
+where $D$ is diffusion constant. Unlike velocity autocorrelation
-+
+function which is averaging over time origins and over all the
-+
+atoms, dipole autocorrelation are calculated for the entire system.
-+
+The dipole autocorrelation function is given by:
-+
+\[
-+
+c_{dipole}  = \left\langle {u_{tot} (t) \cdot u_{tot} (t)}
-+
+\right\rangle
-+
+\]
-+
+Here $u_{tot}$ is the net dipole of the entire system and is given
-+
+by
-+
+\[
-+
+u_{tot} (t) = \sum\limits_i {u_i (t)}
-+
+\]
-+
+In principle, many time correlation functions can be related with
-+
+Fourier transforms of the infrared, Raman, and inelastic neutron
-+
+scattering spectra of molecular liquids. In practice, one can
-+
+extract the IR spectrum from the intensity of dipole fluctuation at
-+
+each frequency using the following relationship:
-+
+\[
-+
+\hat c_{dipole} (v) = \int_{ - \infty }^\infty  {c_{dipole} (t)e^{ -
-+
+i2\pi vt} dt}
-+
+\]
-<
+Some relatively important transformation,
->
+\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
->
->
+Rigid bodies are frequently involved in the modeling of different
->
+areas, from engineering, physics, to chemistry. For example,
->
+missiles and vehicle are usually modeled by rigid bodies.  The
->
+movement of the objects in 3D gaming engine or other physics
->
+simulator is governed by the rigid body dynamics. In molecular
->
+simulation, rigid body is used to simplify the model in
->
+protein-protein docking study\cite{Gray2003}.
->
->
+It is very important to develop stable and efficient methods to
->
+integrate the equations of motion of orientational degrees of
->
+freedom. Euler angles are the nature choice to describe the
->
+rotational degrees of freedom. However, due to its singularity, the
->
+numerical integration of corresponding equations of motion is very
->
+inefficient and inaccurate. Although an alternative integrator using
->
+different sets of Euler angles can overcome this
->
+difficulty\cite{Barojas1973}, the computational penalty and the lost
->
+of angular momentum conservation still remain. A singularity free
->
+representation utilizing quaternions was developed by Evans in
->
+\cite{Evans1977}. Unfortunately, this approach suffer from the
->
+nonseparable Hamiltonian resulted from quaternion representation,
->
+which prevents the symplectic algorithm to be utilized. Another
->
+different approach is to apply holonomic constraints to the atoms
->
+belonging to the rigid body. Each atom moves independently under the
->
+normal forces deriving from potential energy and constraint forces
->
+which are used to guarantee the rigidness. However, due to their
->
+iterative nature, SHAKE and Rattle algorithm converge very slowly
->
+when the number of constraint increases\cite{Ryckaert1977,
->
+Andersen1983}.
->
->
+The break through in geometric literature suggests that, in order to
->
+develop a long-term integration scheme, one should preserve the
->
+symplectic structure of the flow. Introducing conjugate momentum to
->
+rotation matrix $Q$ and re-formulating Hamiltonian's equation, a
->
+symplectic integrator, RSHAKE\cite{Kol1997}, was proposed to evolve
->
+the Hamiltonian system in a constraint manifold by iteratively
->
+satisfying the orthogonality constraint $Q_T Q = 1$. An alternative
->
+method using quaternion representation was developed by
->
+Omelyan\cite{Omelyan1998}. However, both of these methods are
->
+iterative and inefficient. In this section, we will present a
->
+symplectic Lie-Poisson integrator for rigid body developed by
->
+Dullweber and his coworkers\cite{Dullweber1997} in depth.
->
->
+\subsection{\label{introSection:constrainedHamiltonianRB}Constrained Hamiltonian for Rigid Body}
->
+The motion of the rigid body is Hamiltonian with the Hamiltonian
->
+function
->
+\begin{equation}
->
+H = \frac{1}{2}(p^T m^{ - 1} p) + \frac{1}{2}tr(PJ^{ - 1} P) +
->
+V(q,Q) + \frac{1}{2}tr[(QQ^T  - 1)\Lambda ].
->
+\label{introEquation:RBHamiltonian}
->
+\end{equation}
->
+Here, $q$ and $Q$  are the position and rotation matrix for the
->
+rigid-body, $p$ and $P$  are conjugate momenta to $q$  and $Q$ , and
->
+$J$, a diagonal matrix, is defined by
+\[
-<
+L(\cos at) = \frac{p}{{p^2  + a^2 }}
->
+I_{ii}^{ - 1}  = \frac{1}{2}\sum\limits_{i \ne j} {J_{jj}^{ - 1} }
+\]
-+
+where $I_{ii}$ is the diagonal element of the inertia tensor. This
-+
+constrained Hamiltonian equation subjects to a holonomic constraint,
-+
+\begin{equation}
-+
+Q^T Q = 1, \label{introEquation:orthogonalConstraint}
-+
+\end{equation}
-+
+which is used to ensure rotation matrix's orthogonality.
-+
+Differentiating \ref{introEquation:orthogonalConstraint} and using
-+
+Equation \ref{introEquation:RBMotionMomentum}, one may obtain,
-+
+\begin{equation}
-+
+Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0 . \\
-+
+\label{introEquation:RBFirstOrderConstraint}
-+
+\end{equation}
-+
+Using Equation (\ref{introEquation:motionHamiltonianCoordinate},
-+
+\ref{introEquation:motionHamiltonianMomentum}), one can write down
-+
+the equations of motion,
+\[
-<
+L(\sin at) = \frac{a}{{p^2  + a^2 }}
->
+\begin{array}{c}
->
+ \frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\
->
+ \frac{{dp}}{{dt}} =  - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\
->
+ \frac{{dQ}}{{dt}} = PJ^{ - 1}  \label{introEquation:RBMotionRotation}\\
->
+ \frac{{dP}}{{dt}} =  - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\
->
+ \end{array}
+\]
-+
+In general, there are two ways to satisfy the holonomic constraints.
-+
+We can use constraint force provided by lagrange multiplier on the
-+
+normal manifold to keep the motion on constraint space. Or we can
-+
+simply evolve the system in constraint manifold. These two methods
-+
+are proved to be equivalent. The holonomic constraint and equations
-+
+of motions define a constraint manifold for rigid body
+\[
-<
+L(1) = \frac{1}{p}
->
+M = \left\{ {(Q,P):Q^T Q = 1,Q^T PJ^{ - 1}  + J^{ - 1} P^T Q = 0}
->
+\right\}.
+\]
-<
+First, the bath coordinates,
->
+Unfortunately, this constraint manifold is not the cotangent bundle
->
+$T_{\star}SO(3)$. However, it turns out that under symplectic
->
+transformation, the cotangent space and the phase space are
->
+diffeomorphic. Introducing
+\[
-<
+p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) =  - \omega
-<
+_\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha
-<
+}}L(x)
->
+\tilde Q = Q,\tilde P = \frac{1}{2}\left( {P - QP^T Q} \right),
+\]
-+
+the mechanical system subject to a holonomic constraint manifold $M$
-+
+can be re-formulated as a Hamiltonian system on the cotangent space
+\[
-<
+L(x_\alpha  ) = \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) +
-<
+px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }}
->
+T^* SO(3) = \left\{ {(\tilde Q,\tilde P):\tilde Q^T \tilde Q =
->
+,\tilde Q^T \tilde PJ^{ - 1}  + J^{ - 1} P^T \tilde Q = 0} \right\}
->
+\]
->
->
+For a body fixed vector $X_i$ with respect to the center of mass of
->
+the rigid body, its corresponding lab fixed vector $X_0^{lab}$  is
->
+given as
->
+\begin{equation}
->
+X_i^{lab} = Q X_i + q.
->
+\end{equation}
->
+Therefore, potential energy $V(q,Q)$ is defined by
->
+\[
->
+V(q,Q) = V(Q X_0 + q).
->
+\]
->
+Hence, the force and torque are given by
->
+\[
->
+\nabla _q V(q,Q) = F(q,Q) = \sum\limits_i {F_i (q,Q)},
->
+\]
->
+and
->
+\[
->
+\nabla _Q V(q,Q) = F(q,Q)X_i^t
->
+\]
->
+respectively.
->
->
+As a common choice to describe the rotation dynamics of the rigid
->
+body, angular momentum on body frame $\Pi  = Q^t P$ is introduced to
->
+rewrite the equations of motion,
->
+\begin{equation}
->
+ \begin{array}{l}
->
+ \mathop \Pi \limits^ \bullet   = J^{ - 1} \Pi ^T \Pi  + Q^T \sum\limits_i {F_i (q,Q)X_i^T }  - \Lambda  \\
->
+ \mathop Q\limits^{{\rm{   }} \bullet }  = Q\Pi {\rm{ }}J^{ - 1}  \\
->
+ \end{array}
->
+ \label{introEqaution:RBMotionPI}
->
+\end{equation}
->
+, as well as holonomic constraints,
->
+\[
->
+\begin{array}{l}
->
+ \Pi J^{ - 1}  + J^{ - 1} \Pi ^t  = 0 \\
->
+ Q^T Q = 1 \\
->
+ \end{array}
->
+\]
->
->
+For a vector $v(v_1 ,v_2 ,v_3 ) \in R^3$ and a matrix $\hat v \in
->
+so(3)^ \star$, the hat-map isomorphism,
->
+\begin{equation}
->
+v(v_1 ,v_2 ,v_3 ) \Leftrightarrow \hat v = \left(
->
+{\begin{array}{*{20}c}
->
+& { - v_3 } & {v_2 }  \\
->
+   {v_3 } & 0 & { - v_1 }  \\
->
+   { - v_2 } & {v_1 } & 0  \\
->
+\end{array}} \right),
->
+\label{introEquation:hatmapIsomorphism}
->
+\end{equation}
->
+will let us associate the matrix products with traditional vector
->
+operations
->
+\[
->
+\hat vu = v \times u
->
+\]
->
->
+Using \ref{introEqaution:RBMotionPI}, one can construct a skew
->
+matrix,
->
+\begin{equation}
->
+(\mathop \Pi \limits^ \bullet   - \mathop \Pi \limits^ \bullet  ^T
->
+){\rm{ }} = {\rm{ }}(\Pi  - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi  + \Pi J^{
->
+- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T  - X_i F_i (r,Q)^T Q]} -
->
+(\Lambda  - \Lambda ^T ) . \label{introEquation:skewMatrixPI}
->
+\end{equation}
->
+Since $\Lambda$ is symmetric, the last term of Equation
->
+\ref{introEquation:skewMatrixPI} is zero, which implies the Lagrange
->
+multiplier $\Lambda$ is absent from the equations of motion. This
->
+unique property eliminate the requirement of iterations which can
->
+not be avoided in other methods\cite{Kol1997, Omelyan1998}.
->
->
+Applying hat-map isomorphism, we obtain the equation of motion for
->
+angular momentum on body frame
->
+\begin{equation}
->
+\dot \pi  = \pi  \times I^{ - 1} \pi  + \sum\limits_i {\left( {Q^T
->
+F_i (r,Q)} \right) \times X_i }.
->
+\label{introEquation:bodyAngularMotion}
->
+\end{equation}
->
+In the same manner, the equation of motion for rotation matrix is
->
+given by
->
+\[
->
+\dot Q = Qskew(I^{ - 1} \pi )
->
+\]
->
->
+\subsection{\label{introSection:SymplecticFreeRB}Symplectic
->
+Lie-Poisson Integrator for Free Rigid Body}
->
->
+If there is not external forces exerted on the rigid body, the only
->
+contribution to the rotational is from the kinetic potential (the
->
+first term of \ref{ introEquation:bodyAngularMotion}). The free
->
+rigid body is an example of Lie-Poisson system with Hamiltonian
->
+function
->
+\begin{equation}
->
+T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 )
->
+\label{introEquation:rotationalKineticRB}
->
+\end{equation}
->
+where $T_i^r (\pi _i ) = \frac{{\pi _i ^2 }}{{2I_i }}$ and
->
+Lie-Poisson structure matrix,
->
+\begin{equation}
->
+J(\pi ) = \left( {\begin{array}{*{20}c}
->
+& {\pi _3 } & { - \pi _2 }  \\
->
+   { - \pi _3 } & 0 & {\pi _1 }  \\
->
+   {\pi _2 } & { - \pi _1 } & 0  \\
->
+\end{array}} \right)
->
+\end{equation}
->
+Thus, the dynamics of free rigid body is governed by
->
+\begin{equation}
->
+\frac{d}{{dt}}\pi  = J(\pi )\nabla _\pi  T^r (\pi )
->
+\end{equation}
->
->
+One may notice that each $T_i^r$ in Equation
->
+\ref{introEquation:rotationalKineticRB} can be solved exactly. For
->
+instance, the equations of motion due to $T_1^r$ are given by
->
+\begin{equation}
->
+\frac{d}{{dt}}\pi  = R_1 \pi ,\frac{d}{{dt}}Q = QR_1
->
+\label{introEqaution:RBMotionSingleTerm}
->
+\end{equation}
->
+where
->
+\[ R_1  = \left( {\begin{array}{*{20}c}
->
+& 0 & 0  \\
->
+& 0 & {\pi _1 }  \\
->
+& { - \pi _1 } & 0  \\
->
+\end{array}} \right).
->
+\]
->
+The solutions of Equation \ref{introEqaution:RBMotionSingleTerm} is
->
+\[
->
+\pi (\Delta t) = e^{\Delta tR_1 } \pi (0),Q(\Delta t) =
->
+Q(0)e^{\Delta tR_1 }
->
+\]
->
+with
->
+\[
->
+e^{\Delta tR_1 }  = \left( {\begin{array}{*{20}c}
->
+& 0 & 0  \\
->
+& {\cos \theta _1 } & {\sin \theta _1 }  \\
->
+& { - \sin \theta _1 } & {\cos \theta _1 }  \\
->
+\end{array}} \right),\theta _1  = \frac{{\pi _1 }}{{I_1 }}\Delta t.
+\]
-<
+Then, the system coordinates,
-<
+\begin{align}
-<
+mL(\ddot x) &=  - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} -
-<
+\sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{\frac{{g_\alpha
-<
+}}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha
-<
+(0)}}{{p^2  + \omega _\alpha ^2 }} - \frac{{g_\alpha ^2 }}{{m_\alpha
-<
+}}\omega _\alpha ^2 L(x)} \right\}}
-<
+%
-<
+ &= - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} -
-<
+ \sum\limits_{\alpha  = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}\frac{p}{{p^2  + \omega _\alpha ^2 }}pL(x)
-<
+ - \frac{p}{{p^2  + \omega _\alpha ^2 }}g_\alpha  x_\alpha  (0)
-<
+ - \frac{1}{{p^2  + \omega _\alpha ^2 }}g_\alpha  \dot x_\alpha  (0)} \right\}}
-<
+\end{align}
-<
+Then, the inverse transform,
->
+To reduce the cost of computing expensive functions in $e^{\Delta
->
+tR_1 }$, we can use Cayley transformation,
->
+\[
->
+e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1
->
+)
->
+\]
->
+The flow maps for $T_2^r$ and $T_3^r$ can be found in the same
->
+manner.
-<
+\begin{align}
-<
+m\ddot x &=  - \frac{{\partial W(x)}}{{\partial x}} -
->
+In order to construct a second-order symplectic method, we split the
->
+angular kinetic Hamiltonian function can into five terms
->
+\[
->
+T^r (\pi ) = \frac{1}{2}T_1 ^r (\pi _1 ) + \frac{1}{2}T_2^r (\pi _2
->
+) + T_3^r (\pi _3 ) + \frac{1}{2}T_2^r (\pi _2 ) + \frac{1}{2}T_1 ^r
->
+(\pi _1 )
->
+\].
->
+Concatenating flows corresponding to these five terms, we can obtain
->
+an symplectic integrator,
->
+\[
->
+\varphi _{\Delta t,T^r }  = \varphi _{\Delta t/2,\pi _1 }  \circ
->
+\varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }
->
+\circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi
->
+_1 }.
->
+\]
->
->
+The non-canonical Lie-Poisson bracket ${F, G}$ of two function
->
+$F(\pi )$ and $G(\pi )$ is defined by
->
+\[
->
+\{ F,G\} (\pi ) = [\nabla _\pi  F(\pi )]^T J(\pi )\nabla _\pi  G(\pi
->
+)
->
+\]
->
+If the Poisson bracket of a function $F$ with an arbitrary smooth
->
+function $G$ is zero, $F$ is a \emph{Casimir}, which is the
->
+conserved quantity in Poisson system. We can easily verify that the
->
+norm of the angular momentum, $\parallel \pi
->
+\parallel$, is a \emph{Casimir}. Let$ F(\pi ) = S(\frac{{\parallel
->
+\pi \parallel ^2 }}{2})$ for an arbitrary function $ S:R \to R$ ,
->
+then by the chain rule
->
+\[
->
+\nabla _\pi  F(\pi ) = S'(\frac{{\parallel \pi \parallel ^2
->
+}}{2})\pi
->
+\]
->
+Thus $ [\nabla _\pi  F(\pi )]^T J(\pi ) =  - S'(\frac{{\parallel \pi
->
+\parallel ^2 }}{2})\pi  \times \pi  = 0 $. This explicit
->
+Lie-Poisson integrator is found to be extremely efficient and stable
->
+which can be explained by the fact the small angle approximation is
->
+used and the norm of the angular momentum is conserved.
->
->
+\subsection{\label{introSection:RBHamiltonianSplitting} Hamiltonian
->
+Splitting for Rigid Body}
->
->
+The Hamiltonian of rigid body can be separated in terms of kinetic
->
+energy and potential energy,
->
+\[
->
+H = T(p,\pi ) + V(q,Q)
->
+\]
->
+The equations of motion corresponding to potential energy and
->
+kinetic energy are listed in the below table,
->
+\begin{table}
->
+\caption{Equations of motion due to Potential and Kinetic Energies}
->
+\begin{center}
->
+\begin{tabular}{|l|l|}
->
+  \hline
->
+  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
->
+  Potential & Kinetic \\
->
+  $\frac{{dq}}{{dt}} = \frac{p}{m}$ & $\frac{d}{{dt}}q = p$ \\
->
+  $\frac{d}{{dt}}p =  - \frac{{\partial V}}{{\partial q}}$ & $ \frac{d}{{dt}}p = 0$ \\
->
+  $\frac{d}{{dt}}Q = 0$ & $ \frac{d}{{dt}}Q = Qskew(I^{ - 1} j)$ \\
->
+  $ \frac{d}{{dt}}\pi  = \sum\limits_i {\left( {Q^T F_i (r,Q)} \right) \times X_i }$ & $\frac{d}{{dt}}\pi  = \pi  \times I^{ - 1} \pi$\\
->
+  \hline
->
+\end{tabular}
->
+\end{center}
->
+\end{table}
->
+A second-order symplectic method is now obtained by the
->
+composition of the flow maps,
->
+\[
->
+\varphi _{\Delta t}  = \varphi _{\Delta t/2,V}  \circ \varphi
->
+_{\Delta t,T}  \circ \varphi _{\Delta t/2,V}.
->
+\]
->
+Moreover, $\varphi _{\Delta t/2,V}$ can be divided into two
->
+sub-flows which corresponding to force and torque respectively,
->
+\[
->
+\varphi _{\Delta t/2,V}  = \varphi _{\Delta t/2,F}  \circ \varphi
->
+_{\Delta t/2,\tau }.
->
+\]
->
+Since the associated operators of $\varphi _{\Delta t/2,F} $ and
->
+$\circ \varphi _{\Delta t/2,\tau }$ are commuted, the composition
->
+order inside $\varphi _{\Delta t/2,V}$ does not matter.
->
->
+Furthermore, kinetic potential can be separated to translational
->
+kinetic term, $T^t (p)$, and rotational kinetic term, $T^r (\pi )$,
->
+\begin{equation}
->
+T(p,\pi ) =T^t (p) + T^r (\pi ).
->
+\end{equation}
->
+where $ T^t (p) = \frac{1}{2}p^T m^{ - 1} p $ and $T^r (\pi )$ is
->
+defined by \ref{introEquation:rotationalKineticRB}. Therefore, the
->
+corresponding flow maps are given by
->
+\[
->
+\varphi _{\Delta t,T}  = \varphi _{\Delta t,T^t }  \circ \varphi
->
+_{\Delta t,T^r }.
->
+\]
->
+Finally, we obtain the overall symplectic flow maps for free moving
->
+rigid body
->
+\begin{equation}
->
+\begin{array}{c}
->
+ \varphi _{\Delta t}  = \varphi _{\Delta t/2,F}  \circ \varphi _{\Delta t/2,\tau }  \\
->
+  \circ \varphi _{\Delta t,T^t }  \circ \varphi _{\Delta t/2,\pi _1 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t,\pi _3 }  \circ \varphi _{\Delta t/2,\pi _2 }  \circ \varphi _{\Delta t/2,\pi _1 }  \\
->
+  \circ \varphi _{\Delta t/2,\tau }  \circ \varphi _{\Delta t/2,F}  .\\
->
+ \end{array}
->
+\label{introEquation:overallRBFlowMaps}
->
+\end{equation}
->
->
+\section{\label{introSection:langevinDynamics}Langevin Dynamics}
->
+As an alternative to newtonian dynamics, Langevin dynamics, which
->
+mimics a simple heat bath with stochastic and dissipative forces,
->
+has been applied in a variety of studies. This section will review
->
+the theory of Langevin dynamics simulation. A brief derivation of
->
+generalized Langevin equation will be given first. Follow that, we
->
+will discuss the physical meaning of the terms appearing in the
->
+equation as well as the calculation of friction tensor from
->
+hydrodynamics theory.
->
->
+\subsection{\label{introSection:generalizedLangevinDynamics}Derivation of Generalized Langevin Equation}
->
->
+Harmonic bath model, in which an effective set of harmonic
->
+oscillators are used to mimic the effect of a linearly responding
->
+environment, has been widely used in quantum chemistry and
->
+statistical mechanics. One of the successful applications of
->
+Harmonic bath model is the derivation of Deriving Generalized
->
+Langevin Dynamics. Lets consider a system, in which the degree of
->
+freedom $x$ is assumed to couple to the bath linearly, giving a
->
+Hamiltonian of the form
->
+\begin{equation}
->
+H = \frac{{p^2 }}{{2m}} + U(x) + H_B  + \Delta U(x,x_1 , \ldots x_N)
->
+\label{introEquation:bathGLE}.
->
+\end{equation}
->
+Here $p$ is a momentum conjugate to $q$, $m$ is the mass associated
->
+with this degree of freedom, $H_B$ is harmonic bath Hamiltonian,
->
+\[
->
+H_B  = \sum\limits_{\alpha  = 1}^N {\left\{ {\frac{{p_\alpha ^2
->
+}}{{2m_\alpha  }} + \frac{1}{2}m_\alpha  \omega _\alpha ^2 }
->
+\right\}}
->
+\]
->
+where the index $\alpha$ runs over all the bath degrees of freedom,
->
+$\omega _\alpha$ are the harmonic bath frequencies, $m_\alpha$ are
->
+the harmonic bath masses, and $\Delta U$ is bilinear system-bath
->
+coupling,
->
+\[
->
+\Delta U =  - \sum\limits_{\alpha  = 1}^N {g_\alpha  x_\alpha  x}
->
+\]
->
+where $g_\alpha$ are the coupling constants between the bath and the
->
+coordinate $x$. Introducing
->
+\[
->
+W(x) = U(x) - \sum\limits_{\alpha  = 1}^N {\frac{{g_\alpha ^2
->
+}}{{2m_\alpha  w_\alpha ^2 }}} x^2
->
+\] and combining the last two terms in Equation
->
+\ref{introEquation:bathGLE}, we may rewrite the Harmonic bath
->
+Hamiltonian as
->
+\[
->
+H = \frac{{p^2 }}{{2m}} + W(x) + \sum\limits_{\alpha  = 1}^N
->
+{\left\{ {\frac{{p_\alpha ^2 }}{{2m_\alpha  }} + \frac{1}{2}m_\alpha
->
+w_\alpha ^2 \left( {x_\alpha   - \frac{{g_\alpha  }}{{m_\alpha
->
+w_\alpha ^2 }}x} \right)^2 } \right\}}
->
+\]
->
+Since the first two terms of the new Hamiltonian depend only on the
->
+system coordinates, we can get the equations of motion for
->
+Generalized Langevin Dynamics by Hamilton's equations
->
+\ref{introEquation:motionHamiltonianCoordinate,
->
+introEquation:motionHamiltonianMomentum},
->
+\begin{equation}
->
+m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} -
->
+\sum\limits_{\alpha  = 1}^N {g_\alpha  \left( {x_\alpha   -
->
+\frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right)},
->
+\label{introEquation:coorMotionGLE}
->
+\end{equation}
->
+and
->
+\begin{equation}
->
+m\ddot x_\alpha   =  - m_\alpha  w_\alpha ^2 \left( {x_\alpha   -
->
+\frac{{g_\alpha  }}{{m_\alpha  w_\alpha ^2 }}x} \right).
->
+\label{introEquation:bathMotionGLE}
->
+\end{equation}
->
->
+In order to derive an equation for $x$, the dynamics of the bath
->
+variables $x_\alpha$ must be solved exactly first. As an integral
->
+transform which is particularly useful in solving linear ordinary
->
+differential equations, Laplace transform is the appropriate tool to
->
+solve this problem. The basic idea is to transform the difficult
->
+differential equations into simple algebra problems which can be
->
+solved easily. Then applying inverse Laplace transform, also known
->
+as the Bromwich integral, we can retrieve the solutions of the
->
+original problems.
->
->
+Let $f(t)$ be a function defined on $ [0,\infty ) $. The Laplace
->
+transform of f(t) is a new function defined as
->
+\[
->
+L(f(t)) \equiv F(p) = \int_0^\infty  {f(t)e^{ - pt} dt}
->
+\]
->
+where  $p$ is real and  $L$ is called the Laplace Transform
->
+Operator. Below are some important properties of Laplace transform
->
->
+\begin{eqnarray*}
->
+ L(x + y)  & = & L(x) + L(y) \\
->
+ L(ax)     & = & aL(x) \\
->
+ L(\dot x) & = & pL(x) - px(0) \\
->
+ L(\ddot x)& = & p^2 L(x) - px(0) - \dot x(0) \\
->
+ L\left( {\int_0^t {g(t - \tau )h(\tau )d\tau } } \right)& = & G(p)H(p) \\
->
+ \end{eqnarray*}
->
->
->
+Applying Laplace transform to the bath coordinates, we obtain
->
+\begin{eqnarray*}
->
+p^2 L(x_\alpha  ) - px_\alpha  (0) - \dot x_\alpha  (0) & = & - \omega _\alpha ^2 L(x_\alpha  ) + \frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) \\
->
+L(x_\alpha  ) & = & \frac{{\frac{{g_\alpha  }}{{\omega _\alpha  }}L(x) + px_\alpha  (0) + \dot x_\alpha  (0)}}{{p^2  + \omega _\alpha ^2 }} \\
->
+\end{eqnarray*}
->
->
+By the same way, the system coordinates become
->
+\begin{eqnarray*}
->
+ mL(\ddot x) & = & - \frac{1}{p}\frac{{\partial W(x)}}{{\partial x}} \\
->
+  & & \mbox{} - \sum\limits_{\alpha  = 1}^N {\left\{ { - \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}\frac{p}{{p^2  + \omega _\alpha ^2 }}pL(x) - \frac{p}{{p^2  + \omega _\alpha ^2 }}g_\alpha  x_\alpha  (0) - \frac{1}{{p^2  + \omega _\alpha ^2 }}g_\alpha  \dot x_\alpha  (0)} \right\}}  \\
->
+\end{eqnarray*}
->
->
+With the help of some relatively important inverse Laplace
->
+transformations:
->
+\[
->
+\begin{array}{c}
->
+ L(\cos at) = \frac{p}{{p^2  + a^2 }} \\
->
+ L(\sin at) = \frac{a}{{p^2  + a^2 }} \\
->
+ L(1) = \frac{1}{p} \\
->
+ \end{array}
->
+\]
->
+, we obtain
->
+\[
->
+m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} -
+\sum\limits_{\alpha  = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega
+_\alpha  t)\dot x(t - \tau )d\tau  - \left[ {g_\alpha  x_\alpha  (0)
+- \frac{{g_\alpha  }}{{m_\alpha  \omega _\alpha  }}} \right]\cos
+(\omega _\alpha  t) - \frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega
+_\alpha  }}\sin (\omega _\alpha  t)} } \right\}}
-<
+%
-<
+&= - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t
->
+\]
->
+\[
->
+m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t
+{\sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha
+t)\dot x(t - \tau )d} \tau }  + \sum\limits_{\alpha  = 1}^N {\left\{
+\omega _\alpha  }}} \right]\cos (\omega _\alpha  t) +
+\frac{{g_\alpha  \dot x_\alpha  (0)}}{{\omega _\alpha  }}\sin
+(\omega _\alpha  t)} \right\}}
-<
+\end{align}
->
+\]
-+
+Introducing a \emph{dynamic friction kernel}
+\begin{equation}
-+
+\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
-+
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
-+
+\label{introEquation:dynamicFrictionKernelDefinition}
-+
+\end{equation}
-+
+and \emph{a random force}
-+
+\begin{equation}
-+
+R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
-+
+- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
-+
+\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
-+
+(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t),
-+
+\label{introEquation:randomForceDefinition}
-+
+\end{equation}
-+
+the equation of motion can be rewritten as
-+
+\begin{equation}
+m\ddot x =  - \frac{{\partial W}}{{\partial x}} - \int_0^t {\xi
+(t)\dot x(t - \tau )d\tau }  + R(t)
+\label{introEuqation:GeneralizedLangevinDynamics}
+\end{equation}
-<
+%where $ {\xi (t)}$ is friction kernel, $R(t)$ is random force and
-<
+%$W$ is the potential of mean force. $W(x) =  - kT\ln p(x)$
->
+which is known as the \emph{generalized Langevin equation}.
->
->
+\subsubsection{\label{introSection:randomForceDynamicFrictionKernel}Random Force and Dynamic Friction Kernel}
->
->
+One may notice that $R(t)$ depends only on initial conditions, which
->
+implies it is completely deterministic within the context of a
->
+harmonic bath. However, it is easy to verify that $R(t)$ is totally
->
+uncorrelated to $x$ and $\dot x$,
+\[
-<
+\xi (t) = \sum\limits_{\alpha  = 1}^N {\left( { - \frac{{g_\alpha ^2
-<
+}}{{m_\alpha  \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha  t)}
->
+\begin{array}{l}
->
+ \left\langle {x(t)R(t)} \right\rangle  = 0, \\
->
+ \left\langle {\dot x(t)R(t)} \right\rangle  = 0. \\
->
+ \end{array}
+\]
-<
+For an infinite harmonic bath, we can use the spectral density and
-<
+an integral over frequencies.
->
+This property is what we expect from a truly random process. As long
->
+as the model, which is gaussian distribution in general, chosen for
->
+$R(t)$ is a truly random process, the stochastic nature of the GLE
->
+still remains.
-+
+%dynamic friction kernel
-+
+The convolution integral
+\[
-<
+R(t) = \sum\limits_{\alpha  = 1}^N {\left( {g_\alpha  x_\alpha  (0)
-<
+- \frac{{g_\alpha ^2 }}{{m_\alpha  \omega _\alpha ^2 }}x(0)}
-<
+\right)\cos (\omega _\alpha  t)}  + \frac{{\dot x_\alpha
-<
+(0)}}{{\omega _\alpha  }}\sin (\omega _\alpha  t)
->
+\int_0^t {\xi (t)\dot x(t - \tau )d\tau }
+\]
-<
+The random forces depend only on initial conditions.
->
+depends on the entire history of the evolution of $x$, which implies
->
+that the bath retains memory of previous motions. In other words,
->
+the bath requires a finite time to respond to change in the motion
->
+of the system. For a sluggish bath which responds slowly to changes
->
+in the system coordinate, we may regard $\xi(t)$ as a constant
->
+$\xi(t) = \Xi_0$. Hence, the convolution integral becomes
->
+\[
->
+\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = \xi _0 (x(t) - x(0))
->
+\]
->
+and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes
->
+\[
->
+m\ddot x =  - \frac{\partial }{{\partial x}}\left( {W(x) +
->
+\frac{1}{2}\xi _0 (x - x_0 )^2 } \right) + R(t),
->
+\]
->
+which can be used to describe dynamic caging effect. The other
->
+extreme is the bath that responds infinitely quickly to motions in
->
+the system. Thus, $\xi (t)$ can be taken as a $delta$ function in
->
+time:
->
+\[
->
+\xi (t) = 2\xi _0 \delta (t)
->
+\]
->
+Hence, the convolution integral becomes
->
+\[
->
+\int_0^t {\xi (t)\dot x(t - \tau )d\tau }  = 2\xi _0 \int_0^t
->
+{\delta (t)\dot x(t - \tau )d\tau }  = \xi _0 \dot x(t),
->
+\]
->
+and Equation \ref{introEuqation:GeneralizedLangevinDynamics} becomes
->
+\begin{equation}
->
+m\ddot x =  - \frac{{\partial W(x)}}{{\partial x}} - \xi _0 \dot
->
+x(t) + R(t) \label{introEquation:LangevinEquation}
->
+\end{equation}
->
+which is known as the Langevin equation. The static friction
->
+coefficient $\xi _0$ can either be calculated from spectral density
->
+or be determined by Stokes' law for regular shaped particles.A
->
+briefly review on calculating friction tensor for arbitrary shaped
->
+particles is given in Sec.~\ref{introSection:frictionTensor}.
+\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem}
-<
+So we can define a new set of coordinates,
->
->
+Defining a new set of coordinates,
+\[
+q_\alpha  (t) = x_\alpha  (t) - \frac{1}{{m_\alpha  \omega _\alpha
+^2 }}x(0)
-<
+\]
-<
+This makes
->
+\],
->
+we can rewrite $R(T)$ as
+\[
-<
+R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}
->
+R(t) = \sum\limits_{\alpha  = 1}^N {g_\alpha  q_\alpha  (t)}.
+\]
+And since the $q$ coordinates are harmonic oscillators,
-+
-+
+\begin{eqnarray*}
-+
+ \left\langle {q_\alpha ^2 } \right\rangle  & = & \frac{{kT}}{{m_\alpha  \omega _\alpha ^2 }} \\
-+
+ \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle & = & \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
-+
+ \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle & = &\delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
-+
+ \left\langle {R(t)R(0)} \right\rangle & = & \sum\limits_\alpha  {\sum\limits_\beta  {g_\alpha  g_\beta  \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle } }  \\
-+
+  & = &\sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t)}  \\
-+
+  & = &kT\xi (t) \\
-+
+\end{eqnarray*}
-+
-+
+Thus, we recover the \emph{second fluctuation dissipation theorem}
-+
+\begin{equation}
-+
+\xi (t) = \left\langle {R(t)R(0)} \right\rangle
-+
+\label{introEquation:secondFluctuationDissipation}.
-+
+\end{equation}
-+
+In effect, it acts as a constraint on the possible ways in which one
-+
+can model the random force and friction kernel.
-+
-+
+\subsection{\label{introSection:frictionTensor} Friction Tensor}
-+
+Theoretically, the friction kernel can be determined using velocity
-+
+autocorrelation function. However, this approach become impractical
-+
+when the system become more and more complicate. Instead, various
-+
+approaches based on hydrodynamics have been developed to calculate
-+
+the friction coefficients. The friction effect is isotropic in
-+
+Equation, $\zeta$ can be taken as a scalar. In general, friction
-+
+tensor $\Xi$ is a $6\times 6$ matrix given by
+\[
-+
+\Xi  = \left( {\begin{array}{*{20}c}
-+
+   {\Xi _{}^{tt} } & {\Xi _{}^{rt} }  \\
-+
+   {\Xi _{}^{tr} } & {\Xi _{}^{rr} }  \\
-+
+\end{array}} \right).
-+
+\]
-+
+Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction
-+
+tensor and rotational resistance (friction) tensor respectively,
-+
+while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $
-+
+{\Xi^{rt} }$ is rotation-translation coupling tensor. When a
-+
+particle moves in a fluid, it may experience friction force or
-+
+torque along the opposite direction of the velocity or angular
-+
+velocity,
-+
+\[
-+
+\left( \begin{array}{l}
-+
+ F_R  \\
-+
+ \tau _R  \\
-+
+ \end{array} \right) =  - \left( {\begin{array}{*{20}c}
-+
+   {\Xi ^{tt} } & {\Xi ^{rt} }  \\
-+
+   {\Xi ^{tr} } & {\Xi ^{rr} }  \\
-+
+\end{array}} \right)\left( \begin{array}{l}
-+
+ v \\
-+
+ w \\
-+
+ \end{array} \right)
-+
+\]
-+
+where $F_r$ is the friction force and $\tau _R$ is the friction
-+
+toque.
-+
-+
+\subsubsection{\label{introSection:resistanceTensorRegular}The Resistance Tensor for Regular Shape}
-+
-+
+For a spherical particle, the translational and rotational friction
-+
+constant can be calculated from Stoke's law,
-+
+\[
-+
+\Xi ^{tt}  = \left( {\begin{array}{*{20}c}
-+
+   {6\pi \eta R} & 0 & 0  \\
-+
+& {6\pi \eta R} & 0  \\
-+
+& 0 & {6\pi \eta R}  \\
-+
+\end{array}} \right)
-+
+\]
-+
+and
-+
+\[
-+
+\Xi ^{rr}  = \left( {\begin{array}{*{20}c}
-+
+   {8\pi \eta R^3 } & 0 & 0  \\
-+
+& {8\pi \eta R^3 } & 0  \\
-+
+& 0 & {8\pi \eta R^3 }  \\
-+
+\end{array}} \right)
-+
+\]
-+
+where $\eta$ is the viscosity of the solvent and $R$ is the
-+
+hydrodynamics radius.
-+
-+
+Other non-spherical shape, such as cylinder and ellipsoid
-+
+\textit{etc}, are widely used as reference for developing new
-+
+hydrodynamics theory, because their properties can be calculated
-+
+exactly. In 1936, Perrin extended Stokes's law to general ellipsoid,
-+
+also called a triaxial ellipsoid, which is given in Cartesian
-+
+coordinates by\cite{Perrin1934, Perrin1936}
-+
+\[
-+
+\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2
-+
+}} = 1
-+
+\]
-+
+where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately,
-+
+due to the complexity of the elliptic integral, only the ellipsoid
-+
+with the restriction of two axes having to be equal, \textit{i.e.}
-+
+prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved
-+
+exactly. Introducing an elliptic integral parameter $S$ for prolate,
-+
+\[
-+
+S = \frac{2}{{\sqrt {a^2  - b^2 } }}\ln \frac{{a + \sqrt {a^2  - b^2
-+
+} }}{b},
-+
+\]
-+
+and oblate,
-+
+\[
-+
+S = \frac{2}{{\sqrt {b^2  - a^2 } }}arctg\frac{{\sqrt {b^2  - a^2 }
-+
+}}{a}
-+
+\],
-+
+one can write down the translational and rotational resistance
-+
+tensors
-+
+\[
+\begin{array}{l}
-<
+ \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  = \left\langle {q_\alpha ^2 (0)} \right\rangle \cos (\omega _\alpha  t) \\
-<
+ \left\langle {q_\alpha  (t)q_\beta  (0)} \right\rangle  = \delta _{\alpha \beta } \left\langle {q_\alpha  (t)q_\alpha  (0)} \right\rangle  \\
-<
+ \end{array}
->
+ \Xi _a^{tt}  = 16\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - b^2 )S - 2a}} \\
->
+ \Xi _b^{tt}  = \Xi _c^{tt}  = 32\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - 3b^2 )S + 2a}} \\
->
+ \end{array},
+\]
-+
+and
-+
+\[
-+
+\begin{array}{l}
-+
+ \Xi _a^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^2  - b^2 )b^2 }}{{2a - b^2 S}} \\
-+
+ \Xi _b^{rr}  = \Xi _c^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^4  - b^4 )}}{{(2a^2  - b^2 )S - 2a}} \\
-+
+ \end{array}.
-+
+\]
-<
+\begin{align}
-<
+\left\langle {R(t)R(0)} \right\rangle  &= \sum\limits_\alpha
-<
+{\sum\limits_\beta  {g_\alpha  g_\beta  \left\langle {q_\alpha
-<
+(t)q_\beta  (0)} \right\rangle } }
-<
+%
-<
+&= \sum\limits_\alpha  {g_\alpha ^2 \left\langle {q_\alpha ^2 (0)}
-<
+\right\rangle \cos (\omega _\alpha  t)}
-<
+%
-<
+&= kT\xi (t)
-<
+\end{align}
->
+\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}The Resistance Tensor for Arbitrary Shape}
-+
+Unlike spherical and other regular shaped molecules, there is not
-+
+analytical solution for friction tensor of any arbitrary shaped
-+
+rigid molecules. The ellipsoid of revolution model and general
-+
+triaxial ellipsoid model have been used to approximate the
-+
+hydrodynamic properties of rigid bodies. However, since the mapping
-+
+from all possible ellipsoidal space, $r$-space, to all possible
-+
+combination of rotational diffusion coefficients, $D$-space is not
-+
+unique\cite{Wegener1979} as well as the intrinsic coupling between
-+
+translational and rotational motion of rigid body, general ellipsoid
-+
+is not always suitable for modeling arbitrarily shaped rigid
-+
+molecule. A number of studies have been devoted to determine the
-+
+friction tensor for irregularly shaped rigid bodies using more
-+
+advanced method where the molecule of interest was modeled by
-+
+combinations of spheres(beads)\cite{Carrasco1999} and the
-+
+hydrodynamics properties of the molecule can be calculated using the
-+
+hydrodynamic interaction tensor. Let us consider a rigid assembly of
-+
+$N$ beads immersed in a continuous medium. Due to hydrodynamics
-+
+interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different
-+
+than its unperturbed velocity $v_i$,
-+
+\[
-+
+v'_i  = v_i  - \sum\limits_{j \ne i} {T_{ij} F_j }
-+
+\]
-+
+where $F_i$ is the frictional force, and $T_{ij}$ is the
-+
+hydrodynamic interaction tensor. The friction force of $i$th bead is
-+
+proportional to its ``net'' velocity
+\begin{equation}
-<
+\xi (t) = \left\langle {R(t)R(0)} \right\rangle
-<
+\label{introEquation:secondFluctuationDissipation}
->
+F_i  = \zeta _i v_i  - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }.
->
+\label{introEquation:tensorExpression}
+\end{equation}
-+
+This equation is the basis for deriving the hydrodynamic tensor. In
-+
+, Oseen and Burgers gave a simple solution to Equation
-+
+\ref{introEquation:tensorExpression}
-+
+\begin{equation}
-+
+T_{ij}  = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij}
-+
+R_{ij}^T }}{{R_{ij}^2 }}} \right).
-+
+\label{introEquation:oseenTensor}
-+
+\end{equation}
-+
+Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$.
-+
+A second order expression for element of different size was
-+
+introduced by Rotne and Prager\cite{Rotne1969} and improved by
-+
+Garc\'{i}a de la Torre and Bloomfield\cite{Torre1977},
-+
+\begin{equation}
-+
+T_{ij}  = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I +
-+
+\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma
-+
+_i^2  + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} -
-+
+\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right].
-+
+\label{introEquation:RPTensorNonOverlapped}
-+
+\end{equation}
-+
+Both of the Equation \ref{introEquation:oseenTensor} and Equation
-+
+\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij}
-+
+\ge \sigma _i  + \sigma _j$. An alternative expression for
-+
+overlapping beads with the same radius, $\sigma$, is given by
-+
+\begin{equation}
-+
+T_{ij}  = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 -
-+
+\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I +
-+
+\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right]
-+
+\label{introEquation:RPTensorOverlapped}
-+
+\end{equation}
-<
+\section{\label{introSection:hydroynamics}Hydrodynamics}
->
+To calculate the resistance tensor at an arbitrary origin $O$, we
->
+construct a $3N \times 3N$ matrix consisting of $N \times N$
->
+$B_{ij}$ blocks
->
+\begin{equation}
->
+B = \left( {\begin{array}{*{20}c}
->
+   {B_{11} } &  \ldots  & {B_{1N} }  \\
->
+    \vdots  &  \ddots  &  \vdots   \\
->
+   {B_{N1} } &  \cdots  & {B_{NN} }  \\
->
+\end{array}} \right),
->
+\end{equation}
->
+where $B_{ij}$ is given by
->
+\[
->
+B_{ij}  = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij}
->
+)T_{ij}
->
+\]
->
+where $\delta _{ij}$ is Kronecker delta function. Inverting matrix
->
+$B$, we obtain
-<
+\subsection{\label{introSection:frictionTensor} Friction Tensor}
-<
+\subsection{\label{introSection:analyticalApproach}Analytical
-<
+Approach}
->
+\[
->
+C = B^{ - 1}  = \left( {\begin{array}{*{20}c}
->
+   {C_{11} } &  \ldots  & {C_{1N} }  \\
->
+    \vdots  &  \ddots  &  \vdots   \\
->
+   {C_{N1} } &  \cdots  & {C_{NN} }  \\
->
+\end{array}} \right)
->
+\]
->
+, which can be partitioned into $N \times N$ $3 \times 3$ block
->
+$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$
->
+\[
->
+U_i  = \left( {\begin{array}{*{20}c}
->
+& { - z_i } & {y_i }  \\
->
+   {z_i } & 0 & { - x_i }  \\
->
+   { - y_i } & {x_i } & 0  \\
->
+\end{array}} \right)
->
+\]
->
+where $x_i$, $y_i$, $z_i$ are the components of the vector joining
->
+bead $i$ and origin $O$. Hence, the elements of resistance tensor at
->
+arbitrary origin $O$ can be written as
->
+\begin{equation}
->
+\begin{array}{l}
->
+ \Xi _{}^{tt}  = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\
->
+ \Xi _{}^{tr}  = \Xi _{}^{rt}  = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\
->
+ \Xi _{}^{rr}  =  - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j  \\
->
+ \end{array}
->
+\label{introEquation:ResistanceTensorArbitraryOrigin}
->
+\end{equation}
-<
+\subsection{\label{introSection:approximationApproach}Approximation
-<
+Approach}
->
+The resistance tensor depends on the origin to which they refer. The
->
+proper location for applying friction force is the center of
->
+resistance (reaction), at which the trace of rotational resistance
->
+tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of
->
+resistance is defined as an unique point of the rigid body at which
->
+the translation-rotation coupling tensor are symmetric,
->
+\begin{equation}
->
+\Xi^{tr}  = \left( {\Xi^{tr} } \right)^T
->
+\label{introEquation:definitionCR}
->
+\end{equation}
->
+Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin},
->
+we can easily find out that the translational resistance tensor is
->
+origin independent, while the rotational resistance tensor and
->
+translation-rotation coupling resistance tensor depend on the
->
+origin. Given resistance tensor at an arbitrary origin $O$, and a
->
+vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can
->
+obtain the resistance tensor at $P$ by
->
+\begin{equation}
->
+\begin{array}{l}
->
+ \Xi _P^{tt}  = \Xi _O^{tt}  \\
->
+ \Xi _P^{tr}  = \Xi _P^{rt}  = \Xi _O^{tr}  - U_{OP} \Xi _O^{tt}  \\
->
+ \Xi _P^{rr}  = \Xi _O^{rr}  - U_{OP} \Xi _O^{tt} U_{OP}  + \Xi _O^{tr} U_{OP}  - U_{OP} \Xi _O^{tr} ^{^T }  \\
->
+ \end{array}
->
+ \label{introEquation:resistanceTensorTransformation}
->
+\end{equation}
->
+where
->
+\[
->
+U_{OP}  = \left( {\begin{array}{*{20}c}
->
+& { - z_{OP} } & {y_{OP} }  \\
->
+   {z_i } & 0 & { - x_{OP} }  \\
->
+   { - y_{OP} } & {x_{OP} } & 0  \\
->
+\end{array}} \right)
->
+\]
->
+Using Equations \ref{introEquation:definitionCR} and
->
+\ref{introEquation:resistanceTensorTransformation}, one can locate
->
+the position of center of resistance,
->
+\begin{eqnarray*}
->
+ \left( \begin{array}{l}
->
+ x_{OR}  \\
->
+ y_{OR}  \\
->
+ z_{OR}  \\
->
+ \end{array} \right) & = &\left( {\begin{array}{*{20}c}
->
+   {(\Xi _O^{rr} )_{yy}  + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} }  \\
->
+   { - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz}  + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} }  \\
->
+   { - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx}  + (\Xi _O^{rr} )_{yy} }  \\
->
+\end{array}} \right)^{ - 1}  \\
->
+  & & \left( \begin{array}{l}
->
+ (\Xi _O^{tr} )_{yz}  - (\Xi _O^{tr} )_{zy}  \\
->
+ (\Xi _O^{tr} )_{zx}  - (\Xi _O^{tr} )_{xz}  \\
->
+ (\Xi _O^{tr} )_{xy}  - (\Xi _O^{tr} )_{yx}  \\
->
+ \end{array} \right) \\
->
+\end{eqnarray*}
-<
+\subsection{\label{introSection:centersRigidBody}Centers of Rigid
-<
+Body}
->
->
->
+where $x_OR$, $y_OR$, $z_OR$ are the components of the vector
->
+joining center of resistance $R$ and origin $O$.

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+Removed lines
-+
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->
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Comparing trunk/tengDissertation/Introduction.tex (file contents): Revision 2703 by tim, Wed Apr 12 04:13:16 2006 UTC vs. Revision 2793 by tim, Mon Jun 5 21:24:52 2006 UTC

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Comparing trunk/tengDissertation/Introduction.tex (file contents):
Revision 2703 by tim, Wed Apr 12 04:13:16 2006 UTC vs.
Revision 2793 by tim, Mon Jun 5 21:24:52 2006 UTC